Hello Bond, without a will, the passing on of all assets will be equal and assuming she hasn't got a husband then the next of kin would fall to you and any siblings.
As you've mentioned in your opening post, the comment regarding the house was mentioned in your brothers presence so best thing to do would be to speak to him about it. If things are amicable and he is still in a comfortable position with his finances and I'm assuming you live in the house at present(?),then hopefully he will be willing to grant your mothers wishes.
It may be worth having it in mind as to your reasons for wanting to act now though, for example if you were looking to cash in and take out all equity to refinance a new house, your brother may be a little put out, despite knowing your mother said she wanted you to have it.
Depending on the value of the house and what equity is available, why not make a suggestion to your brother and still give him something from it? For example, if he has worked and had 100k mortgage whereas you're going to get 100k through inheriting the house, why not say to him you'd like him to have 10-25k to put towards his own mortgage? Using this example, you'll still be walking away with a nice chunk of deposit for your next property.